Problem: Factor the following expression: $5$ $x^2$ $-14$ $x$ $-3$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(-3)} &=& -15 \\ {a} + {b} &=& & & {-14} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-15$ and add them together. Remember, since $-15$ is negative, one of the factors must be negative. The factors that add up to ${-14}$ will be your ${a}$ and ${b}$ When ${a}$ is ${1}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({1})({-15}) &=& -15 \\ {a} + {b} &=& {1} + {-15} &=& -14 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 +{1}x {-15}x {-3} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 +{1}x) + ({-15}x {-3}) $ Factor out the common factors: $ x(5x + 1) - 3(5x + 1) $ Notice how $(5x + 1)$ has become a common factor. Factor this out to find the answer. $(5x + 1)(x - 3)$